# A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality

Daniel Kessler; Ricardo H. Nochetto; Alfred Schmidt

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 38, Issue: 1, page 129-142
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topKessler, Daniel, Nochetto, Ricardo H., and Schmidt, Alfred. "A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality." ESAIM: Mathematical Modelling and Numerical Analysis 38.1 (2010): 129-142. <http://eudml.org/doc/194202>.

@article{Kessler2010,

abstract = {
Phase-field models, the simplest of which is Allen–Cahn's
problem, are characterized by a small parameter ε that dictates
the interface thickness. These models naturally call for mesh adaptation
techniques, which rely on a posteriori error control.
However, their error analysis usually deals with the
underlying non-monotone nonlinearity via a Gronwall argument which
leads to an exponential dependence on ε-2. Using an energy argument
combined with a
topological continuation argument and a spectral estimate, we
establish an a posteriori error control result with only a low order
polynomial dependence in ε-1. Our result is applicable to
any conforming discretization technique that allows for a
posteriori residual estimation. Residual estimators for an
adaptive finite element scheme are derived to illustrate the theory.
},

author = {Kessler, Daniel, Nochetto, Ricardo H., Schmidt, Alfred},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {A posteriori error estimates; phase-field models; adaptive finite element method.; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element},

language = {eng},

month = {3},

number = {1},

pages = {129-142},

publisher = {EDP Sciences},

title = {A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality},

url = {http://eudml.org/doc/194202},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Kessler, Daniel

AU - Nochetto, Ricardo H.

AU - Schmidt, Alfred

TI - A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 1

SP - 129

EP - 142

AB -
Phase-field models, the simplest of which is Allen–Cahn's
problem, are characterized by a small parameter ε that dictates
the interface thickness. These models naturally call for mesh adaptation
techniques, which rely on a posteriori error control.
However, their error analysis usually deals with the
underlying non-monotone nonlinearity via a Gronwall argument which
leads to an exponential dependence on ε-2. Using an energy argument
combined with a
topological continuation argument and a spectral estimate, we
establish an a posteriori error control result with only a low order
polynomial dependence in ε-1. Our result is applicable to
any conforming discretization technique that allows for a
posteriori residual estimation. Residual estimators for an
adaptive finite element scheme are derived to illustrate the theory.

LA - eng

KW - A posteriori error estimates; phase-field models; adaptive finite element method.; Allen-Cahn equation; Phase-field models; mesh adaptation techniques; a posteriori error control; adaptive finite element

UR - http://eudml.org/doc/194202

ER -

## References

top- S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.27 (1979) 1085–1095.
- H. Brézis, Analyse fonctionnelle. Dunod, Paris (1999).
- G. Caginalp and X. Chen, Convergence of the phase-field model to its sharp interface limits. Euro. J. Appl. Math.9 (1998) 417–445.
- X. Chen, Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differantial Equations19 (1994) 1371–1395.
- Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér9 (1975) 77–84.
- R. Dautrey and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Masson (1988).
- P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc.347 (1995) 1533–1589.
- K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems iv: Nonlinear problems. SIAM J. Numer. Anal.32 (1995) 1729–1749.
- X. Feng and A. Prohl, Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Num. Math.94 (2003) 33–65.
- Ch. Makridakis and R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal.41 (2003) 1585–1594.
- J. Rappaz and J.-F. Scheid, Existence of solutions to a phase-field model for the solidification process of a binary alloy. Math. Methods Appl. Sci.23 (2000) 491–513.
- A. Schmidt and K. Siebert, ALBERT: An adaptive hierarchical finite element toolbox. Preprint 06/2000, Freiburg edition.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.